Effects of interventions and optimal strategies in the stochastic system approach to causality

Abstract

We consider the problem of defining the effect of an intervention on a timevarying risk factor or treatment for a disease or a physiological marker; we develop here the latter case. So, the system considered is (Y, A, C), where Y = (Yt), is the marker process of interest, A = At the treatment (assumed to take values 0 or 1) and C a potential confounding factor. The marker process Y has a Doob-Meyer decomposition dYt = λtdt + dMt, where the intensity of the process Y , λt is a function of the past history of the three processes and can be written as φ( Ȳt−, Āt−, C)), where Xt means the information of X up to time t; the function φ(•, •, •) is the "physical law" and cannot be changed. Y lives in continuous time but can be observed only at discrete times by: Zj = Yt j + εj. A realistic case is that the treatment can be changed only at discrete times, according to a probability law: P(At j = 1| Zj, Āt j−1 , C). In an observation study the treatment attribution law is unknown; however, the physical law can be estimated without knowing the treatment attribution law, provided a well specified model is available. An intervention is specified by the treatment attribution law, which is thus known. Simple interventions will simply randomize the attribution of the treatment; interventions that take into account the past history will be called "strategies". The effect of interventions can be defined by a risk function R int = Eint[L( Ȳt J , Āt J , C)], where L( Ȳt J , Āt J , C) is a loss function, and contrasts between risk functions for different strategies can be formed. Simple contrasts between two strategies, like Eint1(Yt J ) − Eint0(Yt J ), are very particular cases of this approach. Once we can compute effects for any strategy, we can search for optimal or sub-optimal strategies; in particular we can find optimal parametric strategies. We present several ways for designing strategies. As an illustration, we consider the choice of a strategy for containing the HIV load below a certain level while limiting the treatment burden. A simulation study demonstrates the possibility of finding optimal parametric strategies.